Linear control system controllability

In a linear system of the form $$\dot{x}=Ax+Bu$$ and $$y=Cx$$ we can use state feedback control $$u=Kx$$ assuming we know the state, or can observe it, to control the system if (A,B) is controllable (all states are reachable).

How does the theory on controllability apply to output feedback? So that the feedback is not taken from the state of the system using an observer, but from the output.

If the original system (A,B) is not controllable, is it also true that no OUTPUT feedback law exists to control the system from any initial condition?

My initial reaction is that the system is controllable from neither the state feedback or output feedback methods. My basis for this assumption is that the theory of controllability is fundamentally based on the system (A,B), and is not developed from the starting point of any feedback. But I am unsure.

If the original system (A,B) is not controllable, is it also true that no OUTPUT feedback law exists to control the system from any initial condition?

yes.

Think about it this way. the outputs are simply the C matrix multiplied by the states. If you cannot control what the states are, you cannot control what the output is. That means with feedback there is still no way you can control the output of all of the states.

note... you can still have partial control-ability and system level stability.

is this a real life problem, or a schoolwork/textbook problem. Is it purely theoretical or is there a real system?
If you 'need' to solve this problem, you can look into modifying the system such that it is no longer uncontrolable